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The exercise you've provided involves a bounded sequence and a function. Let's break it down step by step:

### 1. Problem Setup:

- You are given a sequence \((u_n)\), where \(u_0 = 0\), and for all \(n \in \mathbb{N}\), the recurrence relation is defined as:
  \[
  u_{n+1} = \frac{u_n + 3}{4u_n + 4}
  \]
  
- There is also a function \( f \) defined on the interval \( ]-1, +\infty[ \) by:
  \[
  f(x) = \frac{x + 3}{4x + 4}
  \]

### 2. Tasks:

1. **Étudier les variations de \(f\)**:  
   You need to study the variations (monotonicity) of the function \(f(x)\), which means you have to find the derivative \(f'(x)\) and determine where the function is increasing or decreasing.

2. **Démontrer par récurrence que pour tout entier naturel \(n\), \( 0 \leq u_n \leq 1\)**:  
   This asks you to prove by mathematical induction that for all natural numbers \(n\), the sequence \((u_n)\) stays within the bounds \( 0 \leq u_n \leq 1 \).

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### **Solution Outline:**

#### 1. Study the Variations of \( f(x) \)

The function is:
\[
f(x) = \frac{x + 3}{4x + 4}
\]
To study the variations, we compute its derivative:

- First, apply the quotient rule to \( f(x) = \frac{x + 3}{4x + 4} \).

The quotient rule is:  
\[
f'(x) = \frac{(4x + 4) \cdot 1 - (x + 3) \cdot 4}{(4x + 4)^2}
\]
Simplifying the numerator and denominator will give us the derivative \(f'(x)\).

#### 2. Prove by Induction that \(0 \leq u_n \leq 1\)

To prove by induction:

- **Base Case**: Verify for \(n = 0\).  
  We know that \( u_0 = 0 \), and \(0 \leq 0 \leq 1\), so the base case holds.

- **Inductive Step**:  
  Assume that \( 0 \leq u_n \leq 1 \) holds for some \(n \geq 0\). You need to show that this implies \( 0 \leq u_{n+1} \leq 1 \).

  Using the recurrence relation:
  \[
  u_{n+1} = \frac{u_n + 3}{4u_n + 4}
  \]
  We can analyze the behavior of this function within the bounds of \( 0 \leq u_n \leq 1 \) to prove the step.

Would you like me to proceed with solving both steps in detail? Let me know if you'd like further clarification!

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Here are 5 related questions that might interest you:

1. How do we determine the critical points of a function using its derivative?
2. What does the monotonicity of a function tell us about its behavior?
3. How does the recurrence relation help in analyzing sequences?
4. Can the fixed points of a function be used to analyze the long-term behavior of a sequence?
5. What role does mathematical induction play in proving properties of sequences?

**Tip**: When applying the quotient rule, remember to carefully simplify the numerator to avoid errors in sign or expression.

Soit la suite (u_n) définie par u_0 = 0 et pour tout entier naturel n, u_{n+1} = (u_n + 3) / (4u_n + 4). Étudier les variations de f(x) = (x + 3) / (4x + 4) et démontrer par récurrence que pour tout entier naturel n, 0 ≤ u_n ≤ 1.

Analyze a bounded sequence defined by recurrence relations and a function f(x). This exercise covers monotonicity via derivative analysis and proving sequence bounds using mathematical induction. Key topics include the quotient rule and recurrence relations. Suitable for advanced high school or undergraduate students.

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